Confinement in a Higgs model on R3×S1

We determine the phase structure of an SU(2) gauge theory with an adjoint scalar on R3×S1 using semiclassical methods. There are two global symmetries: a Z(2) _H symmetry associated with the Higgs field and a Z(2) _C center symmetry associated with the Polyakov loop in the compact direction. The order of the deconfining phase transition can be either second-order or first-order for SU(2), depending on the deformation used. After finding order parameters for the global symmetries, we show that there are four distinct phases: a deconfined phase, a confined phase, a Higgs phase, and a mixed confined phase. The mixed confined phase occurs where one might expect a phase in which there is both confinement and the Higgs mechanism, but the behavior of the order parameters distinguishes the two phases. In the mixed confined phase, the Z(2) _C×Z(2) _H global symmetry breaks spontaneously to a Z(2) subgroup that acts nontrivially on both the scalar field and the Polyakov loop. We find explicitly the BPS and KK monopole solutions of the Euclidean field equations in the BPS limit; these monopoles are extensions of similar pure gauge theory solutions, where they are constituents of instantons. In the mixed phase, a linear combination of the Higgs field φ and A ₄, the component of the gauge field in the compact direction, enters into the monopole solutions. In all four phases, Wilson loops orthogonal to the compact direction are expected to show area-law behavior. We show that this confining behavior can be attributed to a dilute monopole gas in a broad region that includes portions of all four phases. The dilute monopole gas picture breaks down when the action of a BPS monopole is zero. A duality argument similar to that applied recently to the Seiberg-Witten model on R3×S1 shows that the monopole gas picture, arrived at using Euclidean instanton methods, can be interpreted as a gas of finite-energy dyons.